diff --git a/spin_one_periodic_haldane_v2.txt b/spin_one_periodic_haldane_v2.txt
index f93564e..f5547b9 100644
--- a/spin_one_periodic_haldane_v2.txt
+++ b/spin_one_periodic_haldane_v2.txt
@@ -96,7 +96,7 @@ spatial weights doubles the length.  Exact cancellation identities
 couple the two improvements (Lemma~lem:rounded-update), so
 starting from two finite purity bounds both defects decay doubly
 exponentially as length and inverse temperature double together
-(Proposition~prop:bootstrap).  Moment inequalities fill in
+(Proposition~prop:bootstrap).  Moment ineualities fill in
 every intermediate even length.  Finally, purity of the physical
 Gibbs distribution bounds each excited-to-ground weight ratio, hence
 the energy gap.
@@ -123,8 +123,8 @@ given interaction.
 
 The Affleck--Kennedy--Lieb--Tasaki construction gave a rigorous
 gapped spin-one chain with biquadratic interactions
-[cite].  Its bond interaction is h+tfrac13h^2 up
-to a scalar, with h=mathbf S_jbfS_j·mathbf S_bfS_{j+1}, whereas
+[cite].  Its bond interaction is h+1/3h^2 up
+to a scalar, with h=S_j·S_{j+1}, whereas
 (eq:main-H) uses h itself.  Yarotsky proved that a unique
 ground state and a uniform
 periodic gap persist under sufficiently small translation-invariant
@@ -275,7 +275,7 @@ For every nge4 and gin SO(3), -(3n)/(2)Ile H_n(g)le nI.
 Proof. 
 The upper bound is hle I plus unitary conjugation of the seam.  For
 the lower bound consider T=-h_{12}-h_{23} on three open sites in
-Cartesian coordinates.  By (eq:cartesian-bond) an unequal pair
+Cartesian coordinates.  By (eq:cartesian-bond) an uneual pair
 swaps with coefficient -1, an equal pair changes to either other equal
 pair with coefficient 1, and there are no diagonal terms.  Give each
 three-letter word a positive weight q according to its equality
@@ -298,7 +298,7 @@ An open triad containing the seam is unitarily equivalent to an
 untwisted triad (delete the seam bond and rotate all sites on one side,
 using rotation invariance of the internal interaction), so the bound
 holds for every consecutive pair of bonds of H_n(g).  Summing the n
-triad inequalities counts each bond twice and gives
+triad ineualities counts each bond twice and gives
 H_n(g)ge-(3n)/(2)I.
 
 
@@ -352,7 +352,7 @@ these labeled simplexes, with Lebesgue measure on each and unit mass on
 the empty history.  The measure is finite:
 
  mu(Ω_b)=sum_{kge0}(3^kb^k)/(k!)=e^{3b},
-  K_b=L^2(Ω_b,calK_b=L^2(Ω_b,mu;C).
+  K_b=L^2(Ω_b,K_b=L^2(Ω_b,mu;C).
 
 For two histories x,y, merge their labeled times increasingly and let
 k_b(x,y) be the trace of the resulting time-ordered product of the
@@ -488,7 +488,7 @@ even nge4,
  S(n,b)=frac{Z_{2n}(b)}{Z_n(b)^2},
  T(n,b)=(Z_n(2b))/(Z_n(b)^2).
 
-The shift aL cancels from both ratios.  First, the inequality behind
+The shift aL cancels from both ratios.  First, the ineuality behind
 their improvement under squaring.
 
 [Squaring probabilities]
@@ -587,7 +587,7 @@ Define S,T by (eq:purities) and G by
 (eq:bootstrap-coefficient).  Let n_0ge4 even, b_0>0, and let
 u_0,C satisfy
 
- 0<u_0<tfrac16, G(u_0)le C, r:=Cu_0<1, C(1-3u_0)>3,
+ 0<u_0<1/6, G(u_0)le C, r:=Cu_0<1, C(1-3u_0)>3,
 
 with seeds
 
@@ -635,7 +635,7 @@ which combine to
 The bracket lies in (0,1] and the exponent in [1/2,1].  For
 n=n_j, b=b_j:
 
- T(L,b_j)ge(1-u_j)^3ge1-3u_j>tfrac12
+ T(L,b_j)ge(1-u_j)^3ge1-3u_j>1/2
  (n_jle Lle2n_j, L even).
 
 
@@ -683,7 +683,7 @@ So the two rows give the bounds (4/105)log(80/79) and (log20)/784
 of (eq:bootstrap-gap), once their seeds hold.  The second row will
 come from six applications of Lemma~lem:rounded-update to bounds
 based at (36,49/4).  Everything left is finite: establish the two seed
-inequalities without losing the margins these updates need.
+ineualities without losing the margins these updates need.
 
 # Two finite initializations
 
@@ -753,9 +753,9 @@ Two elementary lemmas convert the tables into spectral bounds.
 Let (t_i) be finite or countable with 0le t_ile h and
 sum_it_ile m, where h>0, mge0.  For rge1 put
 
- C(m,h,r)=qh^r+(m-qh)^r,calC(m,h,r)=qh^r+(m-qh)^r, q=floor( m/h).
+ C(m,h,r)=qh^r+(m-qh)^r,C(m,h,r)=qh^r+(m-qh)^r, q=floor( m/h).
 
-Then sum_it_i^rleC(m,h,r);calC(m,h,r); without the cap,
+Then sum_it_i^rleC(m,h,r);C(m,h,r); without the cap,
 sum_it_i^rle m^r.
 
 
@@ -822,7 +822,7 @@ since the filter at the endpoint exceeds its bound by more than
 0.0301 (I) and 0.3463 (N), and each filter increases with |x|
 beyond its endpoint (c_1>0, c_0+c_1x^2>0 there).
 
-The inequalities U^{10}<m_I<2U^{10} and V^{10}<m_N<2V^{10} allow the
+The ineualities U^{10}<m_I<2U^{10} and V^{10}<m_N<2V^{10} allow the
 two-piece (eq:capped-function).  For k=30,120 set
 [
  A_k=U^k+(m_I-U^{10})^{k/10}, B_k=m_O^{k/10},
@@ -846,7 +846,7 @@ The scalar estimates are
  (D+11E)/(12)letau, (D+3E)/(4)letau_0,
 
 with positive margins exceeding 0.001068, 0.000433,
-0.0000472 in the first, second and fourth strict inequalities; all
+0.0000472 in the first, second and fourth strict ineualities; all
 numbers are rational and stated above.
 
 Raise the inverse temperature to b=(5/2)b_0.  For positive Boltzmann
@@ -960,13 +960,13 @@ v_J and every N eigenvalue below v_N.
 The twelfth-moment upper bounds are m_J=1.0657205, m_N=0.2783155.
 Put ell=999/1000 and, for k=36,72,
 
- B(k)=3C(m_N,v_N^calC(m_N,v_N^{12},k/12),
+ B(k)=3C(m_N,v_N^C(m_N,v_N^{12},k/12),
  R_A(k)=B(k)+(m_J-ell^{12})^{k/12}
 
 (with m_J>ell^{12}).  If all J eigenvalues had modulus at most
 ell, Lemma~lem:cappedmass would give
 Z_{72}(b_0)leC(m_J,ell^{12},6)+B(72)<1, with strict margin
-above 0.0693; the trial inequality contradicts this.  So some
+above 0.0693; the trial ineuality contradicts this.  So some
 lambda_0 in J has |lambda_0|>ell; removing it leaves twelfth
 moment at most m_J-ell^{12} in J, and with the three N sectors
 R_A(k) bounds the remaining full kth moment.  Thus
@@ -1052,7 +1052,7 @@ and non-scalar, so the bootstrap criterion applies.
 Proposition~prop:seed60 gives
 S(60,105/4)>7/8 and T(120,105/4)>7/8.  Apply
 Proposition~prop:bootstrap with
-(n_0,b_0,u_0,C)=(60,105/4,1/8,79/10); its parameter inequalities hold
+(n_0,b_0,u_0,C)=(60,105/4,1/8,79/10); its parameter ineualities hold
 by Section~sec:bootstrap, and Cu_0=79/80.  Hence the ground
 state is unique and gamma_L>frac4{105}log(80/79) for every even
 Lge60, proving (eq:gap60).
@@ -1084,7 +1084,7 @@ cite[arXiv v4, End Matter, Definition~11 and Theorem~12]{Tasaki2025}.
 A local operator on the spin chain Z acts on finitely many sites; a
 state omega is a linear functional on local operators with
 omega(I)=1, omega(A^*A)ge0.  Put
-[H,A]=sum_{jinZ}[mathbf S_jbfS_j·mathbf S_bfS_{j+1},A], only finitely
+[H,A]=sum_{jinZ}[S_j·S_{j+1},A], only finitely
 many terms nonzero.
 
 
@@ -1106,7 +1106,7 @@ local operators; the compatible limits form a positive normalized
 functional omega.  For Lge2304, Theorem~thm:main gives a
 unique ground state psi_L and gap above delta=log(20)/784.
 Writing omega_L(A)=<psi_L,Apsi_L>, the spectral
-inequality on the orthogonal complement of psi_L gives
+ineuality on the orthogonal complement of psi_L gives
 [
  omega_L(A^*[H_L,A])gedelta(omega_L(A^*A)-|omega_L(A)|^2).
 ]
@@ -1315,16 +1315,16 @@ D_{w,w}=32-3n-2sum_{i=0}^{n-1}w_{i-1}w_i, an entry -2 for an allowed
 opposite unit step at letters i-1,i, and +2 on the twisted seam
 i=0 (indices mod n).  Lemma~lem:energy gives
 [
- 0letfrac32nI+Hletfrac52nI,
- spec(R)⊆[1-5n/32,1]⊆[-tfrac78,1],
+ 0le3/2nI+Hle5/2nI,
+ spec(R)⊆[1-5n/32,1]⊆[-7/8,1],
 ]
 so R is a self-adjoint contraction and
 [
- F=exp![-tfrac b2(tfrac32nI+H)]=exp[98(R-I)] .
+ F=exp![-tfrac b2(3/2nI+H)]=exp[98(R-I)] .
 ]
 On the full physical space (direct sum over weight blocks),
 
- Z_n(b,g)=e^{x_n}norm{F}_F^2, x_n=bn(tfrac32-a).
+ Z_n(b,g)=e^{x_n}norm{F}_F^2, x_n=bn(3/2-a).
 
 Define the conditioned Poisson probabilities
 
@@ -1346,7 +1346,7 @@ then e<4.
 
 For a unit basis column e_r, start at X=0 and for j=J,...,0 do
 
- X←Biglfloor((DX)/(32)Bigr)+c_je_r,
+ X←Big⌊((DX)/(32))+c_je_r,
  c_j=floor( Qp_j),
 
 with coordinatewise floor; all quantities are integers, and since
@@ -1458,7 +1458,7 @@ table, direct evaluation yields
  L_{n,g}>frac{C_{n,g}-30}{10^6},
  U_{n,g}<frac{C_{n,g}+30}{10^6}(4le nle12, g=I,P),
 ]
-each strict inequality with margin above 13/10^6.  This gives a
+each strict ineuality with margin above 13/10^6.  This gives a
 reproducible finite verification of all eighteen enclosures and proves the
 b=49/4 table.